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Time-Dependent Fields of a Current-Carrying Wire Time-dependent fields of a current-carrying wire D V Redˇzi´c1 and V Hnizdo2 1Faculty of Physics, University of Belgrade, PO Box 44, 11000 Beograd, Serbia 2National Institute for Occupational Safety and Health, Morgantown, WV 26505, USA E-mail: [email protected] Abstract. The electric and magnetic fields of an infinite straight wire carrying a steady current which is turned on abruptly are determined using Jefimenko’s equations, starting from the standard assumption that the wire is electrically neutral in its rest frame. Some nontrivial aspects of the solution are discussed in detail. 1. Introduction Consider an infinite straight linear wire carrying the current I(t) = 0 for t 0, ≤ and I(t) = I0 for t > 0. That is, a constant current I0 is turned on abruptly at time t = 0. What are the resulting electric and magnetic fields? This apparently simple electrodynamic problem is posed and solved as Example 10.2 in the excellent textbook of Griffiths [1]. Starting from the standard assumption that the wire is electrically neutral in its rest frame, without or with the current, which implies that the scalar potential V is zero, the retarded vector potential A is calculated, and then the electric and magnetic fields are obtained according to E = ∂A/∂t and B = ∇ A, respectively. (Note that in this case the Coulomb and Lorenz− gauges lead to same× potentials since the wire is electrically neutral.) While this solution is correct, we believe that the problem has some intriguing aspects and as such deserves further attention. In this note we present a solution to the problem using Jefimenko’s equations and point out some pitfalls which could be dangerous for novices. Thus, hopefully, our analysis should be instructive for advanced undergraduate and beginning graduate students. 2. Solution using retarded potentials For the convenience of the reader, we first give the solution using retarded potentials, arXiv:1301.1573v2 [physics.class-ph] 11 Feb 2013 in some more detail than that given in Griffiths’s book. As is well known, the retarded vector potential A(r,t) at field point r and time t is given by µ0 J (r′,tr) 3 A(r,t)= d r′, (1) 4π r r Z | − ′| where J(r′,tr) is the current density at a source point r′ and the retarded time t = t r r′ /c. Let the infinitely long wire lie along the z axis. The current in the r − | − | Time-dependent fields of a current-carrying wire 2 wire, which is assumed to have an infinitesimal cross section, is turned on abruptly at t = 0, and thus the current density can be expressed as I δ(s) J(s,t)= 0 Θ(t)zˆ, (2) 2π s where s is the distance from the wire, δ(s) is the one-dimensional Dirac delta function ∞ normalized as 0 δ(s) ds =1 and Θ(t) is the Heaviside step function, R 0, if t 0 Θ(t)= (3) 1, if t>≤ 0. The setup is not realistic, but, in principle, it could be realized approximately with a large superconducting loop of negligible cross section in an inhomogeneous axially symmetric magnetic field, the symmetry axis coinciding with the axis of the loop. If the loop, initially at rest with no current, is moved quickly along the symmetry axis into a new resting position, a persistent current is produced in it, since the total magnetic flux through the loop is constant (cf, e.g., [2]). Equations (1) and (2) imply that the vector potential A at a distance s from the wire is given by 2π µ0 ∞ ∞ I0 δ(s′) Θ(t d/c) A(s,t)= zˆ s′ ds′ dφ′ dz′ − 4π 0 0 2π s′ d Z Z Z−∞ 2 2 µ0I0 ∞ Θ(t √s + z′ /c) = zˆ − dz′. (4) 2 2 4π √s + z′ Z−∞ 2 2 2 1 Here, cylindrical coordinates s,φ,z are used and d=[s +s′ 2ss′cos(φ φ′)+(z z′) ]2 , − − − which is the distance between the field point (s,φ,z) and a source point (s′, φ′,z′); in the second line, the integration with respect to s′ and a transformation z z′ z′ 2 2 1/2 − → reduce the distance to (s + z′ ) . As demanded by the problem’s symmetry, the vector potential is independent of z and φ. The Heaviside-function factor in the integrand of the integral in the second line of (4) causes the potential to vanish at times t < s/c and limits the integration interval to the values of z′ satisfying 2 2 2 z′ c t s . (5) | |≤ − Integral (4) for the vectorp potential A thus evaluates as ‡ √c2t2 s2 µ I − dz A(s,t)= 0 0 zˆΘ(t s/c) ′ 2 2 2 2 2 4π − √c t s √s + z′ Z− − µ I = 0 0 zˆ[ln(ct + c2t2 s2) ln s]Θ(t s/c). (6) 2π − − − The electric field is therefore given byp E(s,t)= ∂A(s,t)/∂t − µ I c = 0 0 zˆ Θ(t s/c) (7) − 2π √c2t2 s2 − − The standard convention is understood according to which f(x)Θ(x x0) = 0 whenever x x0, even‡ when the expression f(x) happens not to be defined at these values− of x. ≤ Time-dependent fields of a current-carrying wire 3 and the magnetic field by § B(s,t)= ∇ A(s,t)= (∂A /∂s)φˆ × − z µ I ct = 0 0 φˆ Θ(t s/c). (8) 2πs √c2t2 s2 − − In both (7) and (8), the delta-function terms that arose from the derivatives of the Heaviside step function in (6) dropped out on account of the property f(x)δ(x x )= f(x )δ(x x ) (9) − 0 0 − 0 of the delta function. Inspecting equations (7) and (8) we see that the fields E(s,t) and B(s,t) attain in the limit t their familiar static values 0 and (µ0I0/2πs)φˆ, respectively, and that both these→ fields ∞ diverge when t s/c. → 3. Solution using Jefimenko’s equations As is now well known, starting from the retarded solution to the inhomogeneous wave equations for the fields E and B [4, 5], or from the familiar retarded potentials [1, 6], or otherwise [7], the time-dependent generalizations of the Coulomb and Biot-Savart laws can be derived: ˙ 1 ̺(r′,tr) ̺˙(r′,tr) J(r′,tr) 3 E(r,t)= + d r′, (10) 4πǫ 3 R c 2 RR− c2 0 Z " R R R # ˙ µ0 J(r′,tr) J(r′,tr) 3 B(r,t)= + d r′, (11) 4π 3 c 2 × R Z " R R # where ̺ is the volume charge density, r r , and the dots denote partial R ′ differentiation with respect to time. TheseR equations, ≡ − showing explicitly true sources of E and B, were first derived by Jefimenko [4]. We now shall calculate the fields E and B in the problem at hand using Jefimenko’s equations. In our problem, the charge density vanishes, ̺ =0, (12) since by assumption the wire is electrically neutral, and using equation (2) we get I0 δ(s′) 2 2 J(s′,tr)= Θ(t s + z′ /c) zˆ, (13) 2π s′ − p I δ(s ) ˙ 0 ′ 2 2 J(s′,tr)= Θ(t s/c) δ(t s + z′ /c) zˆ. (14) 2π − s′ − p Here, cylindrical coordinates are used again, and the delta-function property (9) and the taking, with no loss of generality, the field coordinate z to be 0 reduced the retarded time tr to the same value as that in equation (4); the step-function factor in (14) expresses the fact that the step function in (13) entails that not only the It is perhaps worthwhile to note that expression (6) for the vector potential A(s, t) resembles the quasi-static§ vector potential at a distance s from the midpoint of a straight wire of finite length 2 2 2 2l = 2√c t s carrying a constant current I0. A calculation of the magnetic field according to B = ∇ A in− which the distance-dependent length 2l is treated as a constant would be equivalent to the use× of the Biot–Savart law. However, this law is not applicable beyond the quasi-static regime (cf., e.g., [3]). Time-dependent fields of a current-carrying wire 4 current density itself but also its partial time derivative vanishes for times t < s/c. Substitution into Jefimenko’s equation (10) then gives 2 2 I0 ∞ δ(s′) ∞ δ(t √s + z′ /c) E(s,t)= zˆ Θ(t s/c) s′ ds′ dz′ − , (15) 2 2 2 −4πǫ0c − 0 s′ √s + z′ Z Z−∞ which can be evaluated easily using the decomposition of the delta function [5, 8] 2 2 2 c t 2 2 2 2 2 2 δ(t s + z′ /c)= δ(z′ c t s )+ δ(z′ + c t s ) (16) − √c2t2 s2 − − − as p − h p p i µ I c E(s,t)= 0 0 zˆ Θ(t s/c), (17) − 2π √c2t2 s2 − − in full agreement with the electric field (7), obtained using the retarded vector potential. In a similar fashion, using equations (11), (13), (14) and (16) we obtain √c2t2 s2 µ I − dz 1 ∞ δ(t √s2 + z 2/c) B 0 0 φˆ ′ ′ (s,t)= Θ(t s/c)s 2 2 3/2 + − 2 2 dz′ 4π − √c2t2 s2 (s + z′ ) c s + z′ "Z− − Z−∞ # µ I ct = 0 0 φˆ Θ(t s/c), (18) 2πs √c2t2 s2 − − in full agreement with the magnetic field (8), obtained using the retarded vector potential.
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